CONTRACTORS AND FIXED POINTS
3
is a nonlinear operator which has a contractor r and put Fx=x-r(x)Px.
Suppose that the contractor iterative procedure (1.1) converges to
solution x of the operator equation Px=O. But then it is obvious that the
process (1.1) converges to a fixed point x of Fx. Such a fixed point is
called a contractor type fixed point. Thus the Newton-Kantorovich method (3.1)
converges to a contractor type fixed point. Yoshimura [1] has successfully
applied the contractor type fixed point method to the Quantum Field theory.
He has tried a large number of various methods of Functional Analysis including
the Newton-Kantorovich method, and came to the conclusion (see Yoshimura [1] p.
267): "the most powerful method so far available is Altman's method of con-
tractor with non-linear majorant."
Another example of a contractor is given by Mitsui [1] who investigated
the "initial-value adjusting method" for nonlinear boundary value problems.
5. INTEGRAL CONTRACTORS. Consider the nonlinear evolution equation
(5.1) dx/dt=F(t,x),
O~t~T,
x(O)=x0, where the unknown function x is defined
on [Q,bJ with va.lues in a Banach space X, and F:[O,TJxX+X is a continuous
mapping. Then equation (5.1) can be replaced by the following
(5.2) x(t) = x0+
J~F(t,x(s))ds.
Denote by C(O,T;X) the Banach space of all continuous functions from
[Q,TJ into X. Let B(X) denote the Banach space of all bounded linear
operators from X into itself.
Let r:[O,TJxX+B(X) be a bounded continuous mapping such that
II
F(t,x(t) + y(t) +
J~r(s
,x(s) )y(s)ds)
- F(t,x(t)) - r(t,x(t) )y(t)
II
~K
lly(t)
II
for all OtT and some KO. Then we say that F(t,x) has a bounded integral
contracto;
-{!+
Jtn. If, in addition, the integral equation
0
y(t)
+
J~r(s,x(s))y(s)ds=z(t), O~t~T,
has a solution y in C(O,T;X}, for any x and z in C(O,T;X}, then the
integral contractor is said to be regular (see A[lJ and Kuo [1]}.
THEOREM. Suppose F(t,x} has a bounded integral contractor. Then
equation (5.2} has a continuous solution. Moreover, if the bounded integral
contractor is regular, then the solution is unique. (see A[1J, Kuo [1J}.
Kuo [1J has also introduced the notion of a stochastic integral contractor in
order to establish the existence and uniqueness of solutions of stochastic
integral equations.
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